I. INTRODUCTION
Nowadays relativists are interested in theories with more than four dimensional
space-times. Alvarez et al. [1], Randjbar-Daemi et al. [2], Marciano [3] suggested that the
experimental detection of time variation of fundamental constants could provide strong
evidence for the existence of extra dimensions. The extra dimensions in the space-time
contracted to a very small size of Planck length or remain invariant. Further, during
contraction process, extra dimensions produce large amount of entropy which provides
an alternative resolution to the flatness and horizon problem [4]. Misner [5] pointed out
that viscosity plays an important role in the formation of galaxy. Viscosity also accounts
for the large entropy per baryon observed in the present universe [6]. Banerjee et al. [7]
constructed Bianchi type I cosmological models with viscous fluid in higher dimensional
space-time. Chatterjee and Bhui [8] obtained exact solutions of the field equations in a
five dimensional space time with viscous fluid. Singh et al. [9] obtained exact solutions
of the field equations for a five dimensional cosmological model with variable G and bulk
viscosity in Lyra geometry.
The study of string theory is important in the early stages of the evolution of the
universe before the particle creation. Cosmic strings have received considerable attention
in cosmology as they are believed to give rise to density perturbations leading to the
formation of galaxies [10]. Chatterjee [11] constructed massive string cosmological model
in higher dimensional homogeneous space time. Krori et al. [12] constructed Bianchi type
I string cosmological model in higher dimension and obtained that matter and strings
coexist through out the evolution of the universe. Rahaman et al.[13] obtained exact
solutions of the field equations for a five dimensional space time in Lyra Manifold when
the source of gravitation is massive strings.
Lyra [14] modified the Riemannian geometry by introducing a gauge function into
the structure less manifold as a result of which the cosmological constant arises naturally
from the geometry.

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Communications in Physics, Vol. 17, No. 4 (2007), pp. 213-220
HIGHER-DIMENSIONAL STRING COSMOLOGICAL MODEL
WITH BULK VISCOUS FLUID IN LYRA MANIFOLD
G. MOHANTY, G. C. SAMANTA, AND K. L. MAHANTA
Department of Mathematics
Sambalpur University
Jyotivihar-768019, Orissa, INDIA
Abstract. Locally rotationally symmetric five dimensional string cosmological model with bulk
viscous fluid in Lyra manifold is constructed. This model is obtained for time dependent displace-
ment field, i.e. β = β(t) and constant bulk viscous coefficient. Some physical and geometrical
properties of the model are discussed.
I. INTRODUCTION
Nowadays relativists are interested in theories with more than four dimensional
space-times. Alvarez et al. [1], Randjbar-Daemi et al. [2], Marciano [3] suggested that the
experimental detection of time variation of fundamental constants could provide strong
evidence for the existence of extra dimensions. The extra dimensions in the space-time
contracted to a very small size of Planck length or remain invariant. Further, during
contraction process, extra dimensions produce large amount of entropy which provides
an alternative resolution to the flatness and horizon problem [4]. Misner [5] pointed out
that viscosity plays an important role in the formation of galaxy. Viscosity also accounts
for the large entropy per baryon observed in the present universe [6]. Banerjee et al. [7]
constructed Bianchi type I cosmological models with viscous fluid in higher dimensional
space-time. Chatterjee and Bhui [8] obtained exact solutions of the field equations in a
five dimensional space time with viscous fluid. Singh et al. [9] obtained exact solutions
of the field equations for a five dimensional cosmological model with variable G and bulk
viscosity in Lyra geometry.
The study of string theory is important in the early stages of the evolution of the
universe before the particle creation. Cosmic strings have received considerable attention
in cosmology as they are believed to give rise to density perturbations leading to the
formation of galaxies [10]. Chatterjee [11] constructed massive string cosmological model
in higher dimensional homogeneous space time. Krori et al. [12] constructed Bianchi type
I string cosmological model in higher dimension and obtained that matter and strings
coexist through out the evolution of the universe. Rahaman et al.[13] obtained exact
solutions of the field equations for a five dimensional space time in Lyra Manifold when
the source of gravitation is massive strings.
Lyra [14] modified the Riemannian geometry by introducing a gauge function into
the structure less manifold as a result of which the cosmological constant arises naturally
from the geometry.
214 G. MOHANTY, G. C. SAMANTA and K. L. MAHANTA
The analog of Einstein’s field equations based on Lyra’s geometry in normal gauge
as obtained by Sen [15] and Sen and Dunn [16] are
Rij − 12gijR+
3
2
φiφj − 34gijφkφ
k = −χTij (1)
where φi is the displacement vector and other symbols have their usual meanings as in the
Riemannian geometry.
So far the study of string with bulk viscosity in higher dimensional Lyra manifold
is not yet found in the literature. Therefore we have taken an attempt to construct
LRS Bianchi I five dimensional string cosmological model with bulk viscous fluid in Lyra
manifold. Earlier Bali and Upadhaya [17] constructed four dimensional LRS Bianchi type
I string cosmological model with constant bulk viscous coefficient in general relativity. In
this paper the energy momentum tensor is assumed to be the simple extension of usual
four dimensional cases.
II. THE METRIC AND THE FIELD EQUATIONS
Here we consider a five dimensional LRS Bianchi type I metric in the form
ds2 = −dt2 +A2dx2 + B2(dy2 + dz2) + C2dΨ2 (2)
where A, B and C are functions of cosmic time t only.
We assume here that the co-ordinates to be commoving i.e.
u0 = 1 and u1 = u2 = u3 = u4 = 0 (3)
Further we consider here the displacement vector φi in the form
φi = (β(t), 0, 0, 0, 0) (4)
The energy momentum tensor for a cloud of string dust with a bulk viscous fluid
given by Landau and Lifshitz [18], Letelier [19] and Bali and Dave [20] is
Tij = ρuiuj − λwiwj − ξul;l(gij + uiuj), (5)
where ξ is the bulk coefficient of viscosity, ρ is the proper density for a cloud of strings
with particles attached to them , λ is the string tension density, uiis the five velocity of
the particles, gij is the covariant fundamental tensor , wi is the unit space like vector
representing the direction of the string satisfying
uiui = −wiwi = −1 (6)
and
uiwi = 0 (7)
Further the expansion scalar is given by
θ = ul;l (8)
Using equations (4), (5) and (6) the explicit form of field equations (1) for the line
element (2) are obtained as
HIGHER-DIMENSIONAL STRING COSMOLOGICAL MODEL WITH BULK VISCOUS FLUID ... 215
−
(
B′
B
)2
− 2A
′B′
AB
− 2B
′C ′
BC
− A
′C ′
AC
+
3
4
β2 = −χρ (9)
2
B′′
B
+
C ′′
C
+
(
B′
B
)2
+ 2
B′C ′
BC
+
3
4
β2 = χ(λ+ ξθ) (10)
A′′
A
+
B′′
B
+
C ′′
C
+
A′B′
AB
+
B′C ′
BC
+
A′C ′
AC
+
3
4
β2 = χξθ (11)
A′′
A
+ 2
B′′
B
+
(
B′
B
)2
+ 2
A′B′
AB
+
3
4
β2 = χξθ (12)
where dash denotes differentiation with respect to t. In the following section we intend
to derive the exact solutions of the field equations usingβ = β(t) and ξ = ξ0 (con-
stant)(Mohanty and Pattanaik [21], Bali and Upadhaya [17], Bali and Yadhav [22]) in
order to overcome the difficulties due to non linear nature of the field equations.
III. COSMOLOGICAL SOLUTIONS
Here there are six unknowns viz. A, B, C, β, ρ and λ involved in four field equations
(9)-(12). In order to avoid the insufficiency of field equations for solving six unknowns
through four field equations here we consider the power law i.e.
C = Bn (13)
where n is a constant.
Subtracting equation (11) from (12) we find
B′′
B
− C
′′
C
+
(
B′
B
)2
+
A′B′
AB
− A
′C ′
AC
− B
′C ′
BC
= 0 (14)
Substituting equation (13) in equation (14) we get
B′
B
(
B′′
B′
+ (n + 1)
B′
B
+
A′
A
)
= 0, (15)
which yields following three cases
Case I: B
′′
B′ + (n+ 1)
B′
B +
A′
A = 0
Case II: B′ = 0
Case III: B′ = 0 and B
′′
B′ + (n+ 1)
B′
B +
A′
A =0.
Now in the following subsections we intend to derive the exact solutions of the field equa-
tions for the above mentioned cases.
III.1. Case I
B′′
B′
+ (n+ 1)
B′
B
+
A′
A
= 0 (16)
In this case we find
B(t) =
[
(n+ 2)
(∫
dt
A(t)
+ k
)] 1
n+2
216 G. MOHANTY, G. C. SAMANTA and K. L. MAHANTA
from which it is clear given any function A(t)we can find a B(t). Therefore the solutions
are unique. However for further studies here we consider
B′′
B′
+ (n+ 1)
B′
B
= −A
′
A
= k(> 0 cons tan t)
which yields
B =
[
(n+ 2)
(
k1
k
ekt + k2
)] 1
n+2
(17)
and
A = k3e−kt (18)
where k1 6= 0, k2 and k3 6= 0 are constants of integration.
Now equation (13) yields
C =
[
(n+ 2)
(
k1
k
ekt + k2
)] n
n+ 2 . (19)
Substituting equations (17) - (19) in equation (11) we get
3
4
β2 = χξ0
[
−k + k1e
kt
k1
k e
kt + k2
]
+
(2n+ 1)k21e
2kt
(n+ 2)2
(
k1
k e
kt + k2
)2 − k2. (20)
Putting equations (17)-(20) in equations (9) and (10) we find
ρ = kξ0 − k1e
kt
χ
(
k1
k e
kt + k2
)(k + χξ0) (21)
and
λ =
kk1e
kt
χ
(
k1
k e
kt + k2
) − k2
χ
. (22)
The particle density is obtained as
ρp = ρ− λ = kξ0 + k
2
χ
− k1e
kt
χ
(
k1
k e
kt + k2
)(2k+ χξ0). (23)
In this case metric (2) takes the form
ds2 =− dt2 + k23e−2ktdx2
+
[
(n + 2)
(
k1
k
ekt + k2
)] 2
n+2 (
dy2 + dz2
)
+
[
(n + 2)
(
k1
k
ekt + k2
)] 2n
n+2
dΨ2.
(24)
HIGHER-DIMENSIONAL STRING COSMOLOGICAL MODEL WITH BULK VISCOUS FLUID ... 217
III.2. Case II
B′ = 0 (25)
In this case we find
B = a (constant). (26)
Now equation (13) yields
C = an (constant). (27)
Substituting equations (26) and (27) in the field equations (9) - (12) we obtain
3
4
β2 = −χρ (28)
3
3
β2 = χλ+ χξ
A′
A
(29)
A′′
A
+
3
4
β2 = χξ
A′
A
(30)
Here there are four unknowns involved in three equations (28) - (30). In order to get
explicit solutions we have to assume a physical or a mathematical condition. Therefore in
this case we consider the following cases:
ρ+ λ = 0 (Reddy [23, 24]) (31)
ρ = λ (geometric strings) (32)
and
ρ = (1 + ω)λ (Takabayasi string or p− string). (33)
III.2.1. ρ+ λ = 0
Subtracting equation (28) from equation (29) and using relation (31) we obtain
A = constant (34)
Substituting equation (34) in equation (30) we get
β2 = 0 (35)
Therefore in this case the model (20) reduces to empty flat model of Einstein’s theory.
III.2.2. ρ = λ(Letelier [25])
Adding equations (28) and (29) and using relation (32) we find
3
2
β2 = χξ0
A′
A
(36)
Subtracting equation (36) from two times of equation (30) we obtain
A′
A
(
A′′
A′
− χξ0
)
= 0 (37)
which yields
A′ = 0 (38)
218 G. MOHANTY, G. C. SAMANTA and K. L. MAHANTA
or
A′′
A′
− χξ0 = 0 (39)
Using equation (38) we get the same model as obtained in section 3.2.1. However from
equation (39) we obtain
A =
a1
χξ0
eχξ0t + a2 (40)
where a1 and a2 are constants of integration.
Substituting equation (40) in equation (36) we find
3
2
β2 =
χa1e
χξ0t
a1
χξ0
eχξ0t + a2
(41)
Equation (28) with the help of equation (41) yields
ρ(= λ) =
−a1eχξ0t
2
(
a1
χξ0
eχξ0t + a2
) (42)
Now the solutions given by (26), (27), (40), (41) and (42) satisfy the field equations(9)-(12)
only when a1 = 0. This immediately yields
ρ(= λ) = 0
β2 = 0
and
A = constant.
Hence in this case the model is same as obtained earlier in section 3.2.1. Here we mention
that in case II the model for p-string could not be obtained due to highly non linear nature
of the field equations.
III.3. Case III
B′ = 0 and
B′′
B′
+ (n+ 1)
B′
B
+
A′
A
= 0
This case is not acceptable due to indeterminacy of B.
IV. PHYSICAL AND GEOMETRICAL PROPERTIES
In the preceding section the metric (24) represents a string cosmological model with
bulk viscous fluid in Lyra manifold. At the initial epoch t = 0, the metric becomes flat.
As time increases the rate of expansion in the model along y and z axes are faster in
comparison to the contraction of the model along x axes when −2 < n < 0.The extra
dimension contracts if −2 < n < 0.The parameters involved in the model behave as
follows:
(a) The rest energy density for the model (24) given by equation (21) satisfies the
reality condition ρ > 0 when
kξ0χ
(
k1
k
ekt + k2
)
> k1e
kt(k + χξ0).
HIGHER-DIMENSIONAL STRING COSMOLOGICAL MODEL WITH BULK VISCOUS FLUID ... 219
(b) The scalar of expansion for the model is obtained as
θ =
k1e
kt
k1
k e
kt + k2
− k.
At the initial epoch t = 0, θ is finite and θ → 0 when t→∞. The expansion in the
model stops at infinite time. Thus there is finite expansion in the model.
(c) The shear scalar σ2 for the model (24) is
σ2 =
1
2
k21e2kt(n2 + 2)
(n+ 2)2
(
k1
k e
kt + k2
)2 + k1ekt(n+ 1)
(n+ 2)
(
k1
k e
kt + k2
) + k2 + k
.
Since lim
t→∞
σ2
θ2
6= 0 , the universe remains anisotropic throught the evolution.
(d) The spatial volume of the universe is
V =
(n+ 2)
(
k1
k e
kt + k2
)
k3ekt
, k3 > 0
Thus the volume of the universe is finite throught the evolution.
(e) The deceleration parameter (Fienstein et al, [26])
q = −3θ−2
(
θ;iu
i +
1
3
θ2
)
= −
(
3k1ekt
kk2
+ 1
)
.
For k, k1, k2 > 0, the value of the deceleration parameter is negative, which indicates
inflation in the model.
V. CONCLUSION
In this paper we have constructed a five dimensional string cosmological model with
bulk viscous fluid in Lyra manifold. The model obtained is free from initial singularity,
which supports the analysis of Murphy [27] that the introduction of bulk viscous fluid
avoids the initial singularity.
REFERENCES
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Received 06 October 2007.